FZ implies finite derived subgroup: Difference between revisions

Revision as of 22:44, 20 September 2007

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property must also satisfy the second group property
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This result was proved by Schur.

Statement

Verbal statement

If the inner automorphism group (viz the quotient by the center) of a group is finite, so is the commutator subgroup. In fact, there is an explicit bound on the size of the commutator subgroup as a function of the size of the inner automorphism group.

Symbolic statement

Let $G$ be a group such that $I n n (G) = G / Z (G)$ is finite. Then, $G^{'} = [G, G]$ is also finite. In fact, if $| G / Z (G) | = n$ , then $G^{'}$ has size at most $n^{2 n^{3}}$ .

Property-theoretic statement

The group property of being a FZ-group (viz having a finite inner automorphism group) implies the group property of being commutator-finite viz having a finite commutator subgroup.

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 ===Symbolic statement===
-Let <math>G</math> be a [[group]] suchthat <math>Inn(G) = G/Z(G)</math> is finite. Then, <math>G' = [G,G]</math> is also finite. In fact, if <math>|G/Z(G)| = n</math>, then <math>G'</math> has size at most <math>n^{2n^3}</math>.
+Let <math>G</math> be a [[group]] such that <math>Inn(G) = G/Z(G)</math> is finite. Then, <math>G' = [G,G]</math> is also finite. In fact, if <math>|G/Z(G)| = n</math>, then <math>G'</math> has size at most <math>n^{2n^3}</math>.
 ===Property-theoretic statement===
 The group property of being a [[FZ-group]] (viz having a finite inner automorphism group) implies the group property of being [[commutator-finite group|commutator-finite]] viz having a finite [[commutator subgroup]].